Dispersion relations for the hyperbolic thermal conductivity, thermoelasticity and thermoviscoelasticity. (English) Zbl 1392.74032
Summary: The Maxwell-Cattaneo heat conduction theory, the Lord-Shulman theory of thermoelasticity and a hyperbolic theory of thermoviscoelasticity are studied. The dispersion relations are analyzed in the case when a solution is represented in the form of an exponential function decreasing in time. Simple formulas that quite accurately approximate the dispersion curves are obtained. Based on the results of analysis of the dispersion relations, an experimental method of determination of the heat flux relaxation time is suggested.
MSC:
| 74F05 | Thermal effects in solid mechanics |
| 74A15 | Thermodynamics in solid mechanics |
| 80A05 | Foundations of thermodynamics and heat transfer |
| 74D05 | Linear constitutive equations for materials with memory |
Keywords:
hyperbolic thermoelasticity; hyperbolic thermoviscoelasticity; Maxwell-Cattaneo law; heat flux relaxation; dispersion relationsReferences:
| [1] | Pop, E; Sinha, S; Kenneth, E, Goodson heat generation and transport in nanometer-scale transistors, Proc. IEEE, 94, 1587-1601, (2006) · doi:10.1109/JPROC.2006.879794 |
| [2] | Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer, Berlin (1996) · Zbl 0852.73005 · doi:10.1007/978-3-642-97671-1 |
| [3] | Chandrasekharaiah, DS, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev., 51, 705-729, (1998) · doi:10.1115/1.3098984 |
| [4] | Tzou, D.Y.: Macro-to-Microscale Heat Transfer. The Lagging Behaviour. Taylor and Francis, New York (1997) |
| [5] | Shashkov, A. G., Bubnov, V. A., Yanovski, S. Y.: Wave phenomena of heat conductivity: system and structural approach. (1993).(in Russian) |
| [6] | Wang, CC, The principle of fading memory, Arch. Ration. Mech. Anal., 18, 343-366, (1965) · Zbl 0125.41601 · doi:10.1007/BF00281325 |
| [7] | Tzou, DY, On the thermal shock wave induced by a moving heat source, Int. J. Heat Mass Transf., 111, 232-238, (1989) |
| [8] | Qiu, TQ; Tien, CL, Short-pulse laser heating on metals, Int. J. Heat Mass Transf., 35, 719-726, (1992) · doi:10.1016/0017-9310(92)90131-B |
| [9] | Sobolev, SL, Transport processes and traveling waves in systems with local nonequilibrium, Sov. Phys. Uspekhi, 34, 217, (1991) · doi:10.1070/PU1991v034n03ABEH002348 |
| [10] | Cattaneo, C, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Compte Rendus, 247, 431-433, (1958) · Zbl 1339.35135 |
| [11] | Vernotte, P, LES paradoxes de la theorie continue de lequation de la chaleur, CR Acad. Sci., 246, 3154-3155, (1958) · Zbl 1341.35086 |
| [12] | Lykov, A. V.: Theory of heat conduction. Vysshaya Shkola, Moscow. (1967): 599. (in russian) |
| [13] | Babenkov, MB; Ivanova, EA, Analysis of the wave propagation processes in heat transfer problems of the hyperbolic type, Contin. Mech. Thermodyn., 26, 483-502, (2014) · Zbl 1341.35162 · doi:10.1007/s00161-013-0315-8 |
| [14] | Peshkov, V, Second sound in helium II, J. Phys., 8, 381, (1944) |
| [15] | Liu, Y; Mandelis, A, Laser optical and photothermal thermometry of solids and thin films, Exp. Methods Phys. Sci., 42, 297-336, (2009) · doi:10.1016/S1079-4042(09)04207-6 |
| [16] | Magunov, AN, Laser thermometry of solids: state of the art and problems, Meas. Tech., 45, 173-181, (2002) · doi:10.1023/A:1015595806622 |
| [17] | Magunov, A.N.: Laser Thermometry of Solids. Cambridge International Science Publishing, Cambridge (2003) |
| [18] | Wang, X.: Experimental Micro/nanoscale Thermal Transport. Wiley, Hoboken (2012) · doi:10.1002/9781118310243 |
| [19] | Krilovich, V.I., Bil, G.N., Ivakin, E.V., Rubanov, A.C.: Experimental determination heat velocity. NASB 1, 129-134 (2000). (in Russian) |
| [20] | Ivakin, EV; Kizak, AI; Rubanov, AS, Active spectroscopy of Rayleigh light-scattering in study of heat-transfer, Izvestiya Akademii Nauk SSSR Seriya fizicheskaya, 56, 130-134, (1992) |
| [21] | Ivakin, EV; Lazaruk, AM; Filipov, VV, Application of laser induced gratings for thermal diffusivity measurements of solids, Proc. SPIE, 2648, 196-206, (1995) · doi:10.1117/12.226167 |
| [22] | Xu, F., Tianjian, Lu: Introduction to Skin Biothermomechanics and Thermal Pain, vol. 7. Science Press, New York (2011) · doi:10.1007/978-3-642-13202-5 |
| [23] | Grassmann, A; Peters, F, Experimental investigation of heat conduction in wet sand, Heat Mass Transf., 35, 289-294, (1999) · doi:10.1007/s002310050326 |
| [24] | Herwig, H; Beckert, K, Experimental evidence about the controversy concerning Fourier or non-Fourier heat conduction in materials with a nonhomogeneous inner structure, Heat Mass Transf., 36, 387-392, (2000) · doi:10.1007/s002310000081 |
| [25] | Kaminski, W, Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure, J. Heat Transf., 112, 555-560, (1990) · doi:10.1115/1.2910422 |
| [26] | Mitra, K; Kumar, S; Vedavarz, A; Moallemi, MK, Experimental evidence of hyperbolic heat conduction in processed meat, J. Heat Transf., 117, 568-573, (1995) · doi:10.1115/1.2822615 |
| [27] | Roetzel, W; Putra, N; Das, Sarit K, Experiment and analysis for non-Fourier conduction in materials with non-homogeneous inner structure, Int. J. Therm. Sci., 42, 541-552, (2003) · doi:10.1016/S1290-0729(03)00020-6 |
| [28] | Vovnenko, NV; Zimin, BA; Sud’enkov, YuV, Nonequilibrium motion of a metal surface exposed to sub-microsecond laser pulses, Zhurnal tekhnicheskoi fiziki, 80, 41-45, (2010) |
| [29] | Sudenkov, YV; Pavlishin, AI, Nanosecond pressure pulses propagating at anomalously high velocities in metal foils, Tech. Phys. Lett., 29, 491-493, (2003) · doi:10.1134/1.1589567 |
| [30] | Szekeres, A; Fekete, B, Continuummechanics-heat conduction-cognition, Period. Polytech. Eng. Mech. Eng., 59, 8, (2015) · doi:10.3311/PPme.7152 |
| [31] | Tzou, DY, An engineering assessment to the relaxation time in thermal wave propagation, Int. J. Heat Mass Transf., 36, 1845-1851, (1993) · Zbl 0772.73007 · doi:10.1016/s0017-9310(05)80171-1 |
| [32] | Gembarovic, J; Majernik, V, Non-Fourier propagation of heat pulses in finite medium, Int. J. Heat Mass Transf., 31, 1073-1080, (1988) · Zbl 0664.73013 · doi:10.1016/0017-9310(88)90095-6 |
| [33] | Poletkin, KV; Gurzadyan, GG; Shang, J; Kulish, V, Ultrafast heat transfer on nanoscale in thin gold films, Appl. Phys. B, 107, 137-143, (2012) · doi:10.1007/s00340-011-4862-z |
| [34] | Sieniutycz, S, The variational principles of classical type for non-coupled non-stationary irreversible transport processes with convective motion and relaxation, S. sieniutycz, Int. J. Heat Mass Transf., 20, 1221-1231, (1977) · Zbl 0378.76062 · doi:10.1016/0017-9310(77)90131-4 |
| [35] | Majumdar, A, Microscale heat conduction in lelectnc thin films, J. Heat Transf., 115, 7, (1993) · doi:10.1115/1.2910673 |
| [36] | Matsunaga, R. H., dos Santos, I.: Measurement of the thermal relaxation time in agar-gelled water. In: Engineering in Medicine and Biology Society (EMBC), 2012 Annual International Conference of the IEEE. IEEE, pp. 5722-5725 (2012) |
| [37] | Lord, H; Shulman, Y, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15, 299-309, (1967) · Zbl 0156.22702 · doi:10.1016/0022-5096(67)90024-5 |
| [38] | Green, AE; Lindsay, KA, Thermoelasticity, J. Elast., 2, 1-7, (1972) · Zbl 0775.73063 · doi:10.1007/BF00045689 |
| [39] | Hetnarski, RB; Ignaczak, J, Solution-like waves in a low-temperature nonlinear thermoelastic solid, Int. J. Eng. Sci., 34, 1767-1787, (1996) · Zbl 0914.73016 · doi:10.1016/S0020-7225(96)00046-8 |
| [40] | Green, AE; Nagdi, PM, Thermoelasticity without energy dissipation, J. Elast., 31, 189-208, (1993) · Zbl 0784.73009 · doi:10.1007/BF00044969 |
| [41] | Ivanova, EA, Derivation of theory of thermoviscoelasticity by means of two-component medium, Acta Mech., 215, 261-286, (2010) · Zbl 1398.74068 · doi:10.1007/s00707-010-0324-7 |
| [42] | Ivanova, EA; Altenbach, H (ed.); Maugin, GA (ed.); Erofeev, V (ed.), On one model of generalised continuum and its thermodynamical interpretation, 151-174, (2011), Berlin · doi:10.1007/978-3-642-19219-7_7 |
| [43] | Ivanova, EA, Derivation of theory of thermoviscoelasticity by means of two-component Cosserat continuum, Tech. Mech., 32, 273-286, (2012) |
| [44] | Ivanova, EA, Description of mechanism of thermal conduction and internal damping by means of two component Cosserat continuum, Acta Mech., 225, 757-795, (2014) · Zbl 1331.74012 · doi:10.1007/s00707-013-0934-y |
| [45] | Niu, T; Dai, W, A hyperbolic two-step model-based finite-difference method for studying thermal deformation in a 3-D thin film exposed to ultrashort pulsed lasers, Numer. Heat Transf. Part A Appl., 53, 1294-1320, (2008) · doi:10.1080/10407780801960118 |
| [46] | Qin, Y.: Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, vol. 184. Springer, Berlin (2008) · Zbl 1173.35004 |
| [47] | Mondal, S., Mallik, S. H., Kanoria, M.: Fractional order two-temperature dual-phase-lag thermoelasticity with variable thermal conductivity. Int. Sch. Res. Not. vol. 2014, Article ID 646049, 13 pages, (2014). doi:10.1155/2014/646049 |
| [48] | Babenkov, MB, Analysis of dispersion relations of a coupled thermoelasticity problem with regard to heat flux relaxation, J. Appl. Mech, Tech. Phys., 52, 941-949, (2011) · Zbl 1298.74061 · doi:10.1134/S0021894411060125 |
| [49] | Babenkov, MB, Propagation of harmonic perturbations in a thermoelastic medium with heat relaxation, J. Appl. Mech. Tech. Phys., 54, 277-286, (2013) · Zbl 1298.74128 · doi:10.1134/S0021894413020132 |
| [50] | Nowacki, W.: Dyn. Probl. Thermoelast. Springer, Berlin (1975) |
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